Exceptional points and pseudo-Hermiticity in real potential scattering

TL;DR

This paper uses a transfer-matrix approach to analyze real potential scattering in two dimensions, revealing how pseudo-Hermiticity and exceptional points influence wave behavior in finite waveguides, bridging quantum mechanics and scattering theory.

Contribution

It introduces a transfer-matrix formulation linked to pseudo-Hermitian Hamiltonians with exceptional points, providing new insights into real potential scattering in two dimensions.

Findings

Exceptional points occur at specific incident wavenumbers.

Spectral properties of pseudo-Hermitian operators determine scattering behavior.

Real and complex eigenvalues influence waveguide scattering phenomena.

Abstract

We employ a recently-developed transfer-matrix formulation of scattering theory in two dimensions to study a class of scattering setups modeled by real potentials. The transfer matrix for these potentials is related to the time-evolution operator for an associated pseudo-Hermitian Hamiltonian operator $\widehat{\mathbf{H}}$ which develops an exceptional point for a discrete set of incident wavenumbers. We use the spectral properties of this operator to determine the transfer matrix of these potentials and solve their scattering problem. We apply our general results to explore the scattering of waves by a waveguide of finite length in two dimensions, where the source of the incident wave and the detectors measuring the scattered wave are positioned at spatial infinities while the interior of the waveguide, which is filled with an inactive material, forms a finite rectangular region of the space. The study of this model allows us to elucidate the physical meaning and implications of the presence of the real and complex eigenvalues of $\widehat{\mathbf{H}}$ and its exceptional points. Our results reveal the relevance of the concepts of pseudo-Hermitian operator and exceptional point in the standard quantum mechanics of closed systems where the potentials are required to be real.